Question

If $BC$ is a diameter of a circle with centre $O$ and $OD$ is perpendicular to the chord $AB$ of a circle, show that $CA=2OD$.

Answer 1

Given : A circle with centre $O$, diameter $BC$ and $OD$ perpendicular to the chord $AB.$

To prove : $CA=2OD$

Proof: Since $OD$ is $\perp$ to chord $AB$
As we know that the perpendicular drawn from the centre to a chord bisects the chord.
So, the point $D$ is bisect the chord $AB$
Thus, $D$ is the midpoint of $AB$.
and also $O$ act as the centre and the mid point of diameter $BC$.

In $\Delta ABC,O$ and $D$ are the mid points of $BC$ and $AB$ respectively
$\Rightarrow OD\parallel ACandOD=\frac{CA}{2}$
(By midpoint theorem, segment joining the mid points of two sides of a triangle is half of the third side)
$\therefore CA=2OD$
Hence, proved.